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While refreshing my memory on details of the Standard Model, I ran into a muddle. I might not exist to ask bizarre questions if the neutron, neutrino, and hydrogen atom were not electrically neutral. Thanks to the cleverly contrived weak hypercharge assignments of the SM, they are.

Weinberg justified these assignments in terms of anomaly cancellation, but the discussion in his book QT of F (1996, v2, p387) assumed the nonexistence of a right-handed neutrino. He found the ratios of hypercharges to be totally constrained. (There were actually two discrete solutions, but one was clearly unphysical.) But nowadays, given the evidence of neutrino mixing, isn’t there one more continuously adjustable hypercharge parameter to worry about? It seems the SM is one constraint short of a unique solution.

Matthew Schwartz’s book QFT and SM (2014, p634) acknowledges the problem but offers three lame excuses for the right-handed neutrino’s orthodox $Y=0$ assignment: nonexistence (?), Majorana character (?), or any other reason (?). I gather that given a different value, the hydrogen atom would still be neutral, but the stability of heavy nuclei would be affected.

Can any of you δημιουργοι offer a better rationale for the values of hypercharge, other than magnetic monopoles, empirical fact, or the anthropic principle? A grand unifying group, perhaps?

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  • $\begingroup$ So you want a theological argument for the provable nonexistence of a zombie GUT in their infernal charnel that would prevent a baseline charge -b for neutrinos, suitably matched by everybody else... Ooof... Not even G-M, R &S persisted there, tied down by a mysterious realism... $\endgroup$ – Cosmas Zachos Mar 1 '18 at 15:15
  • $\begingroup$ Treating hypercharge as yet another diagonal generator of a grand SU(4) besides the two of QCD seems almost as blasphemous as adding Satan to the Holy Trinity. $\endgroup$ – Bert Barrois Mar 4 '18 at 11:22
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Neutrino masses do not prove that there are right-handed neutrinos. The left-handed neutrino can have a Majorana mass, for example by coupling to a SU(2)L triplet Higgs.

The GUT which most compactly gives rise to right-handed neutrinos is SO(10). The hypercharges are underdetermined, but stipulating that the SO(10) Higgses have relatively low-dimensional representations (something implied by string theory, btw) favors the usual hypercharges.

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